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Nontrivial solutions for a boundary value problem with integral boundary conditions

Bingmei Liu1*, Junling Li1 and Lishan Liu2

Author Affiliations

1 College of Sciences, China University of Mining and Technology, Xuzhou, Jiangsu, 221116, China

2 School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong, 273165, China

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Boundary Value Problems 2014, 2014:15  doi:10.1186/1687-2770-2014-15


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/15


Received:27 July 2013
Accepted:13 November 2013
Published:13 January 2014

© 2014 Liu et al.; licensee Springer.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper concerns the existence of nontrivial solutions for a boundary value problem with integral boundary conditions by topological degree theory. Here the nonlinear term is a sign-changing continuous function and may be unbounded from below.

1 Introduction

Consider the following Sturm-Liouville problem with integral boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M6">View MathML</a>, α and β are right continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M7">View MathML</a>, left continuous at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M8">View MathML</a> and nondecreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M9">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M10">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M13">View MathML</a> denote the Riemann-Stieltjes integral of u with respect to α and β, respectively. Here the nonlinear term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M14">View MathML</a> is a continuous sign-changing function and f may be unbounded from below, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M15">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M16">View MathML</a> is continuous and is allowed to be singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M17">View MathML</a>.

Problems with integral boundary conditions arise naturally in thermal conduction problems [1], semiconductor problems [2], hydrodynamic problems [3]. Integral BCs (BCs denotes boundary conditions) cover multi-point BCs and nonlocal BCs as special cases and have attracted great attention, see [4-14] and the references therein. For more information about the general theory of integral equations and their relation with boundary value problems, we refer to the book of Corduneanu [4], Agarwal and O’Regan [5]. Yang [6], Boucherif [8], Chamberlain et al.[10], Feng [11], Jiang et al.[14] focused on the existence of positive solutions for the cases in which the nonlinear term is nonnegative. Although many papers investigated two-point and multi-point boundary value problems with sign-changing nonlinear terms, for example, [15-20], results for boundary value problems with integral boundary conditions when the nonlinear term is sign-changing are rarely seen except for a few special cases [7,12,13].

Inspired by the above papers, the aim of this paper is to establish the existence of nontrivial solutions to BVP (1.1) under weaker conditions. Our findings presented in this paper have the following new features. Firstly, the nonlinear term f of BVP (1.1) is allowed to be sign-changing and unbounded from below. Secondly, the boundary conditions in BVP (1.1) are the Riemann-Stieltjes integral, which includes multi-point boundary conditions in BVPs as special cases. Finally, the main technique used here is the topological degree theory, the first eigenvalue and its positive eigenfunction corresponding to a linear operator. This paper employs different conditions and different methods to solve the same BVP (1.1) as [7]; meanwhile, this paper generalizes the result in [17] to boundary value problems with integral boundary conditions. What we obtain here is different from [6-20].

2 Preliminaries and lemmas

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M18">View MathML</a> be a Banach space with the maximum norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M19">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M20">View MathML</a>. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M21">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M22">View MathML</a>. Then P is a total cone in E, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M23">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M24">View MathML</a> denotes the dual cone of P, namely, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M25">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M26">View MathML</a> denote the dual space of E, then by Riesz representation theorem, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M26">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M28">View MathML</a>

We assume that the following condition holds throughout this paper.

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M29">View MathML</a>is the unique<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M30">View MathML</a>solution of the linear boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M31">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M32">View MathML</a> solve the following inhomogeneous boundary value problems, respectively:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M33">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M37">View MathML</a>.

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M40">View MathML</a>.

Lemma 2.1 ([7])

If (H1) and (H2) hold, then BVP (1.1) is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M41">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M42">View MathML</a>is the Green function for (1.1).

Define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M43">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M44">View MathML</a>

(2.1)

It is easy to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M43">View MathML</a> is a completely continuous nonlinear operator, and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M20">View MathML</a> is a fixed point of A, then u is a solution of BVP (1.1) by Lemma 2.1.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M20">View MathML</a>, define a linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M48">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M49">View MathML</a>

(2.2)

It is easy to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M48">View MathML</a> is a completely continuous nonlinear operator and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M51">View MathML</a> holds. By [7], the spectral radius <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M52">View MathML</a> of K is positive. The Krein-Rutman theorem [21] asserts that there are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M53">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M54">View MathML</a> corresponding to the first eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M55">View MathML</a> of K such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M56">View MathML</a>

(2.3)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M57">View MathML</a>

(2.4)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M58">View MathML</a> is the dual operator of K given by:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M59">View MathML</a>

The representation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M60">View MathML</a>, the continuity of G and the integrability of h imply that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M61">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M62">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M63">View MathML</a>, and (2.4) can be rewritten equivalently as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M64">View MathML</a>

(2.5)

Lemma 2.2 ([7])

If (H1) holds, then there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M65">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M66">View MathML</a>is a subcone ofPand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M67">View MathML</a>.

Lemma 2.3 ([22])

LetEbe a real Banach space and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M68">View MathML</a>be a bounded open set with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M69">View MathML</a>. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M70">View MathML</a>is a completely continuous operator. (1) If there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M71">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M72">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M73">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M74">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M75">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M76">View MathML</a>. (2) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M77">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M78">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M79">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M80">View MathML</a>. Here deg stands for the Leray-Schauder topological degree inE.

Lemma 2.4Assume that (H1), (H2) and the following assumptions are satisfied:

(C1) There exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M54">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M65">View MathML</a>such that (2.3), (2.4) hold andKmapsPinto<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M84">View MathML</a>.

(C2) There exists a continuous operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M85">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M86">View MathML</a>

(C3) There exist a bounded continuous operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M87">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M88">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M89">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M20">View MathML</a>.

(C4) There exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M91">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M92">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M93">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M20">View MathML</a>.

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M95">View MathML</a>, then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M96">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M97">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M98">View MathML</a>.

Proof Choose a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M99">View MathML</a>. From (C2), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M100">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M101">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M102">View MathML</a> implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M103">View MathML</a>

(2.6)

Now we shall show

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M104">View MathML</a>

(2.7)

provided that R is sufficiently large.

In fact, if (2.7) is not true, then there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M105">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M106">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M107">View MathML</a>

(2.8)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M109">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M110">View MathML</a>. Multiply (2.8) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M111">View MathML</a> on both sides and integrate on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M9">View MathML</a>. Then, by (C4), (2.5), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M113">View MathML</a>

(2.9)

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M114">View MathML</a>

(2.10)

By (2.9), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M115">View MathML</a> holds. Then (2.3), (2.6) and (2.10) imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M116">View MathML</a>

(2.11)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M117">View MathML</a> is a constant.

(C3) shows <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M118">View MathML</a> and (C1) implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M119">View MathML</a>. Then (C1), (2.8) and Lemma 2.2 tell us that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M120">View MathML</a>

The definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M84">View MathML</a> yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M122">View MathML</a>

(2.12)

It follows from (2.6), (2.11) and (2.12) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M123">View MathML</a>

(2.13)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M124">View MathML</a> is a constant.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M125">View MathML</a>, then (2.13) deduces that (2.7) holds provided that R is sufficiently large such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M126">View MathML</a>. By (2.13) and Lemma 2.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M127">View MathML</a>

 □

3 Main results

Theorem 3.1Assume that (H1), (H2) hold and the following conditions are satisfied:

(A1) There exist two nonnegative functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M128">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M129">View MathML</a>and one continuous even function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M130">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M131">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M132">View MathML</a>. Moreover, Bis nondecreasing on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M133">View MathML</a>and satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M134">View MathML</a>.

(A2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M135">View MathML</a>is continuous.

(A3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M136">View MathML</a>uniformly on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M137">View MathML</a>.

(A4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M138">View MathML</a>uniformly on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M137">View MathML</a>.

Here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M140">View MathML</a>is the first eigenvalue of the operatorKdefined by (2.2).

Then BVP (1.1) has at least one nontrivial solution.

Proof We first show that all the conditions in Lemma 2.4 are satisfied. By Lemma 2.2, condition (C1) of Lemma 2.4 is satisfied. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M141">View MathML</a> is a continuous operator. By (A1), for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M142">View MathML</a>, there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M143">View MathML</a> such that when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M144">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M145">View MathML</a> holds. Thus, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M146">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M147">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M148">View MathML</a> holds. The fact that B is nondecreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M133">View MathML</a> yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M150">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M151">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M137">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M153">View MathML</a> is an even function, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M150">View MathML</a> holds, which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M157">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M20">View MathML</a>. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M159">View MathML</a>

that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M160">View MathML</a>. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M161">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M20">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M163">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M164">View MathML</a> holds. Therefore H satisfies condition (C2) in Lemma 2.4.

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M165">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M166">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M20">View MathML</a>, then it follows from (A1) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M169">View MathML</a>

which shows that condition (C3) in Lemma 2.4 holds.

By (A3), there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M170">View MathML</a> and a sufficiently large number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M171">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M172">View MathML</a>

(3.1)

Combining (3.1) with (A1), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M173">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M174">View MathML</a>

and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M175">View MathML</a>

(3.2)

Since K is a positive linear operator, from (3.2) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M176">View MathML</a>

So condition (C4) in Lemma 2.4 is satisfied.

According to Lemma 2.4, we derive that there exists a sufficiently large number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M96">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M178">View MathML</a>

(3.3)

From (A4) it follows that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M179">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M180">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M181">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M182">View MathML</a>

(3.4)

Next we will prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M183">View MathML</a>

(3.5)

If there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M184">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M185">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M186">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M187">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M188">View MathML</a> and by (3.4), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M189">View MathML</a>. The nth iteration of this inequality shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M190">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M191">View MathML</a>), so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M192">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M193">View MathML</a>. This yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M194">View MathML</a>, which is a contradictory inequality. Hence, (3.5) holds.

It follows from (3.5) and Lemma 2.3 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M195">View MathML</a>

(3.6)

By (3.3), (3.6) and the additivity of Leray-Schauder degree, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M196">View MathML</a>

So A has at least one fixed point on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M197">View MathML</a>, namely, BVP (1.1) has at least one nontrivial solution. □

Corollary 3.1Using (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M198">View MathML</a>) instead of (A1), the conclusion of Theorem 3.1 remains true.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M198">View MathML</a>) There exist three constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M200">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M201">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M202">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/15/mathml/M203">View MathML</a>

Competing interests

The authors declare that no conflict of interest exists.

Authors’ contributions

All authors participated in drafting, revising and commenting on the manuscript. All authors read and approved the final manuscript.

Acknowledgements

The first two authors were supported financially by the National Natural Science Foundation of China (11201473, 11271364) and the Fundamental Research Funds for the Central Universities (2013QNA35, 2010LKSX09, 2010QNA42). The third author was supported financially by the National Natural Science Foundation of China (11371221, 11071141), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001) and the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province.

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